Massachusetts Institute of Technology
Department of Materials Science and Engineering
77 Massachusetts Avenue, Cambridge MA 02139-4307
3.21 Kinetics of Materials—Spring 2006
May 1, 2006
Lecture 27: Morphological Stability of Moving Interfaces.
References
1. Balluffi, Allen, and Carter, Kinetics of Materials, Chapter 20.
2. Mathematica notebook for Lecture 27
Key Concepts
• Perturbation methods can be applied to investigate the morphological stability of particles growing by
heat flow and by diffusion. Capillarity acts to make simple interface shapes (e.g., planar and spherical
shapes) stable. Both thermal gradients and concentration gradients can act to destabilize the simple
shapes, possibly leading to cellular or dendritic growth morphologies.
• Solidification of a single-component liquid is stable if the liquid is everywhere above its melting
temperature (see KoM Fig. 20.10). If the liquid is undercooled, the solid/liquid interface is unstable
because a small protuberance causes the thermal gradient in the liquid to become steeper, amplifying
the growth of the protuberance (see KoM Fig. 20.11).
• Diffusion-limited growth in an alloy during precipitation is analogous to solidification of an under­
cooled melt, in that a protuberance on the precipitate/matrix interface locally steepens the concentra­
tion gradient in he matrix ahead of the protuberance, tending to amplify its growth.
• In alloy solidification, when the composition of the solid differs from that in the liquid (the usual case),
solute rejection into the liquid ahead of the interface may lead to constitutional supercooling of the
liquid. Melts become constitutionally supercooled when the thermal gradient in the liquid is relatively
shallow, and the solute ”spike” in the liquid results in liquid compositions that are undercooled by
virtue of their solute enrichment (see KoM Figs. 20.4 and 20.12).
• The stability of a spherical precipitate growing by volume diffusion-control has been modeled by
Mullins and Sekerka using the perturbation method. Spherical harmonic functions are used to describe
shape distortions of a sphere (KoM Eq. 20.58; also see Mathematica notebook for this lecture). The
effect of interface curvature on equilibrium concentration at the boundary (Gibbs–Thomson effect) is
included (KoM Eq. 20.61), thus introducing the stabilizing influence of capillarity into the analysis.
A solution to Laplace’s equation (invariant-field approximation) is used to derive an expression for
the diffusion field in the matrix that is in equilibrium with the perturbed sphere, KoM Eq. 20.69). The
expression for the rate of change of the amplitude of the perturbations is given by KoM Eq. 20.73,
and this provides the condition for morphological stability in terms of the “wavelength” n of the
perturbation, the instantaneous size of the growing particle, the concentration gradient G ahead of the
interface, and the capillary stabilizing forces included in the term Γ. The analysis indicates that for
sufficiently large particles there will always be a shape instability (see KoM Eq. 20.77). In practice,
such instabilities are rarely observed in solid/solid precipitation.
Related Exercises in Kinetics of Materials
Review Exercise 20.8, pp. 529–531.